Table Of ContentConcerning the quark condensate
K. Langfeld,1 H. Markum,2 R. Pullirsch,3 C.D. Roberts,4,5 and S.M. Schmidt1,6
1Institut fu¨r Theoretische Physik, Universit¨at Tu¨bingen,
Auf der Morgenstelle 14, D-72076 Tu¨bingen, Germany
2Atominstitut, Technische Universit¨at Wien, A-1040 Vienna, Austria
3Institut fu¨r Theoretische Physik, Universit¨at Regensburg, D-93040 Regensburg, Germany
4Physics Division, Bldg 203, Argonne National Laboratory, Argonne Illinois 60439-4843
5Fachbereich Physik, Universit¨at Rostock, D-18051 Rostock, Germany
6Helmholtz-Gemeinschaft, Ahrstrasse 45, D-53175 Bonn, Germany
A continuum expression for the trace of the massive dressed-quark propagator is used to ex-
plicate a connection between the infrared limit of the QCD Dirac operator’s spectrum and the
quark condensate appearing in the operator product expansion, and the connection is verified via
comparison with a lattice-QCD simulation. The pseudoscalar vacuum polarisation provides agood
3 approximation to thecondensate overa larger range of current-quarkmasses.
0
0 PACSnumbers: 12.38.Aw,11.30.Qc,11.30.Rd,24.85.+p
2
n
I. INTRODUCTION they occur in pairs: {u (x),γ u (x)}, with eigenvalues
a n 5 n
of opposite sign. It follows that in an external gauge
J
8 Dynamicalchiralsymmetrybreaking(DCSB)isacor- field, A, one can write the Green function for a massive
nerstone of hadron physics. This phenomenon whereby, propagating quark in the form
1 even in the absence of a current-quark mass, self-
u (x)u†(y)
v interactions generate a momentum-dependent running S(x,y;A)=hq(x)q¯(y)i = n n , (3)
4 quarkmass,M(p2),thatislargeintheinfrared: M(0)∼ A iλn+m
2 Xn
0.5GeV, but power-law suppressed in the ultraviolet [1]:
0 where m is the current-quark mass and, naturally, the
1
eigenvaluesdependonA. (NB.Theexpectationvaluede-
30 M(p2)larg=e−p2 2π2γm −hq¯qi0 , (1) notes aGrassmannianfunctionalintegralevaluatedwith
0 3 p2 1l(cid:0)n p2 (cid:1) 1−γm a fixed gauge field configuration.) Assuming, e.g., a lat-
/ 2 Λ2QCD tice regularisation,it follows that
h (cid:16) h i(cid:17)
t is impossible in weakly interacting theories. In Eq. (1), 1 2m 1
ucl- γwmith=N1f2/th(3e3n−um2Nbefr)oifsltighhet-mquasasrkanfloamvoaulorsu,sadnidmheq¯nqsii0onis, V ZV d4xhq¯(x)q(x)iA =− V λXn>0 λ2n+m2 , (4)
the renormalisation-group-invariant vacuum quark con-
n where V is the lattice volume [8]. One may now define
densate [2], to which we shallhereafter refer as the OPE
:
v condensate. WhileEq.(1)isexpressedinLandaugauge, a quark condensate as the infinite volume limit of the
Xi hq¯qi0 is gauge parameter independent. In the chiral average in Eq. (4) over all gauge field configurations:
limit the OPE condensate plays a role analogous to that
r 1
a played by the renormalisation-group-invariant current- h0|q¯q|0i:= lim d4x hhq¯(x)q(x)iAi . (5)
quark mass in the massive theory: it sets the scale of V→∞V ZV
the mass function in the ultraviolet. In the infinite volume limit the operator spectrum be-
The evolution of the dressed-quark mass-function in comes dense and Eqs. (4), (5) become
Eq.(1)toalargeandfiniteconstituent-quark-likemassin
the infrared, M(0) ∼ 0.5GeV, is a longstanding predic- ∞ ρ(λ)
−h0|q¯q|0i=2m dλ , (6)
tion of Dyson-Schwingerequation(DSE) studies [3] that λ2+m2
Z0
has recently been confirmed in simulations of quenched
with ρ(λ) the spectral density. This equation expresses
lattice-QCD[4]. AdeterminationoftheOPEcondensate
an assumption that in QCD the full two-point massive-
directly fromlattice-QCD simulations must awaitan ac-
quark Schwinger function, when viewed as a function of
curate chiral extrapolation [5] but DSE models tuned to
reproducemodernlatticedatagive[6](−hq¯qi0)∼Λ3 . the current-quark mass, has a spectral representation.
QCD
It follows formally from Eq. (6) that
Another view of DCSB is obtained by considering the
eigenvaluesandeigenfunctions ofthe masslessEuclidean ∞ ρ(λ) 1
Dirac operator [7]: lim m dλ = πρ(0), (7)
m→0 Z0 λ2+m2 2
γ·Du (x)=iλ u (x). (2)
n n n
and hence one arrives at the chiral limit result
The operatoris anti-Hermitianand hence the eigenfunc-
tions form a complete set, and except for zero modes −h0|q¯q|0i=πρ(0). (8)
2
This is the so-called Banks-Casher relation [10, 11]. It and is obtained subject to the condition that at some
has long been advocated as a means by which a quark large, spacelike ζ2
condensatemaybe measuredinlattice-QCDsimulations
S (p)−1 =iγ·p+m (ζ), (11)
[11] and has been used in analysing chiral symmetry f p2=ζ2 f
restorationatnonzerotemperature[12]andchemicalpo- where m (ζ) is the r(cid:12)enormalised current-quark mass:
tential [13], and to explore the connection between mag- f (cid:12)
netic monopoles and chiral symmetry breaking in U(1) Z (ζ,Λ)m (ζ)=Z (ζ,Λ)mbm(Λ), (12)
4 f 2 f
gauge theory [14]. Much has been learnt [15, 16] by ex-
ploitingthefactthatqualitativefeaturesofthebehaviour with Z4 the renormalisation constant for the scalar part
of ρ(λ) for λ∼0 canbe understood using chiralrandom ofthequarkself-energy. SinceQCDisanasymptotically
matrix theory; i.e., from considerations based solely on free theory, the chiral limit is defined by
QCD’s global symmetries. Z (ζ2,Λ2)mbm(Λ)≡0, Λ≫ζ (13)
Our main goal is to explicate a correspondence be- 2 f
tween the condensate in Eq. (1) and that in Eq. (8). andin this case the scalarprojectionofEq. (9) does not
In Sec. II we discuss the OPE condensate and its con- exhibit an ultraviolet divergence [2, 17].
nection with QCD’s gap equation, and emphasise that Important in describing chiral symmetry is the axial-
the residue of the lowest-mass pole-contribution to the vector Ward-Takahashi identity:
flavour-nonsingletpseudoscalar vacuum polarisation is a
TH TH
direct measure of the OPE condensate [17]. A natu- P ΓH(k;P) = S−1(k )iγ +iγ S−1(k )
ral ability to express DCSB through the formation of µ 5µ + 5 2 5 2 −
a nonzero OPE condensate is fundamental to the suc- −M(ζ)iΓH5 (k;P)−iΓH5 (k;P)M(ζ),
cess of DSE models of hadron phenomena [18]. In Sec. (14)
III we carefully define the trace of the massive dressed-
quark propagatorand use that to illustrate a connection where P is the total momentum entering the vertex.
between ρ(0) and the OPE condensate, which we verify In Eq. (14): M(ζ) = diag[mu(ζ),md(ζ),ms(ζ)], S =
via comparison with a lattice simulation. Section IV is diag[Su,Sd,Ss] and {TH} are flavour matrices, e.g.,
an epilogue. Tπ+ = 1 λ1+iλ2 (we consider SU (3) because chiral
2 f
symmetry is unimportant for heavier quarks); ΓH(k;P)
(cid:0) (cid:1) 5µ
istherenormalisedaxial-vectorvertex,whichisobtained
II. OPE CONDENSATE from the inhomogeneous Bethe-Salpeter equation
TH
A. Gap and Bethe-Salpeter equations ΓH(k;P) =Z γ γ
5µ tu 2 5 µ 2
(cid:20) (cid:21)tu
(cid:2) Λ (cid:3)
DynamicalchiralsymmetrybreakinginQCDisreadily + [χH(q;P)] Krs(q,k;P), (15)
explored using the DSE for the quark self-energy: 5µ sr tu
Zq
S(p)−1 =Z2(iγ·p+mbm) where χH5µ := S(q+)ΓH5µ(q;P)S(q−), q± = q±P/2, and
K(q,k;P)isthefullyrenormalisedquark-antiquarkscat-
Λ λa
+Z g2D (p−q) γ S(q)Γa(q,p), (9) tering kernel; and ΓH is the pseudoscalar vertex,
1 µν 2 µ ν 5
Zq TH
ΓH(k;P) =Z γ
awghaetroeri;n:ΓaνD(µqν;p(k))isistthheerreennoorrmmaalliisseeddddrreesssseedd--qguluaorkn-gplruoopn- (cid:2) 5 Λ (cid:3)tu 4 (cid:20) 5 2 (cid:21)tu
vertex;mbm istheΛ-dependentcurrent-quarkbaremass + χH(q;P) Krs(q,k;P), (16)
thatappearsinthe Lagrangian;and Λ := Λd4q/(2π)4 Zq 5 sr tu
q (cid:2) (cid:3)
represents a translationally-invariant regularisation of with χH := S(q )ΓH(q;P)S(q ). Multiplicative renor-
R R 5 + 5 −
the integral, with Λ the regularisation mass-scale which malisability ensures that no new renormalisation con-
is removed to infinity as the completion of all calcula- stants appear in Eqs. (15) and (16).
tions. The quark-gluon-vertex and quark wave function Flavour-octetpseudoscalarbound statesappearasco-
renormalisation constants, Z1(ζ2,Λ2) and Z2(ζ2,Λ2) re- incident pole solutions of Eqs. (15), (16), namely,
spectively,depend onthe renormalisationpoint, the reg-
1
ularisation mass-scale and the gauge parameter. ΓH(k;P)∝ΓH(k;P)∝ Γ (k;P), (17)
If the current-quark mass changes with flavour, then 5µ 5 P2+m2H H
the solution of Eq. (9) is flavour dependent:
where Γ is the bound state’s Bethe-Salpeter amplitude
H
and m , its mass. (Regular terms are overwhelmed at
S (p)−1 = iγ·pA (p2,ζ2)+B (p2,ζ2) H
f f f the pole.) Consequently, Eq. (14) entails [2, 17]
1
= Zf(p2,ζ2) iγ·p+Mf(p2,ζ2) , (10) fHm2H =rH(ζ)M(Hζ), (18)
(cid:2) (cid:3)
3
where: M(ζ) = tr [M {TH,(TH)t] is the sum of calculated in this way:
H flavour (ζ)
the constituents’ current-quark masses (“t” denotes ma-
trix transpose); and α p2
B (p2)=m 1− ln +... . (24)
f f π m2
f P = Z Λ 1tr TH tγ γ χ (q;P) , (19) " f# !
H µ 2 5 µ H
irH(ζ) = Z4ZqΛ 122trh(cid:0)TH(cid:1)tγ5 χH(q;P) i, (20) Eqduevnaesrrakytemteiarsmssimainpnodtshsheiebnsleecreiineaspniesorntpuzrreobrpoaotvriotainlounteaholefotrotyh.tehOe PcuErrceonnt--
Zq h(cid:0) (cid:1) i
where χ := S(q )Γ (q;P)S(q ) and the expressions
H + H −
are evaluated at P2+m2 =0. B. Pseudoscalar Vacuum Polarisation
H
Equation (19) is the pseudovector projection of the
meson’s Bethe-Salpeter wave function evaluated at the ConsiderthecoloursingletSchwingerfunctiondescrib-
origin in configuration space. It is the precise expres- ing the pseudoscalar vacuum polarisation
sionfortheleptonicdecayconstant. Therenormalisation
1 1
constant, Z2(ζ,Λ), ensures that the r.h.s. is independent ∆5(x)=hq¯(x) λfγ5q(x)q¯(0) λgγ5q(0)i, (25)
of: the regularisation scale, Λ, which may therefore be 2 2
removed to infinity; the renormalisation point; and the
which can be estimated, e.g., in lattice simulations. Its
gauge parameter. Hence it is truly an observable.
renormalised form can completely be expressed in mo-
Equation (20) is the pseudoscalar analogue. Therein
mentum space using quantities introduced already:
the renormalisation constant Z (ζ,Λ) entails that the
4
r.h.s. is independent of the regularisation scale, Λ, and Λ1
the gauge parameter. It also ensures that the ζ- ω5fg(P)=Z22tr 2λfγ5S(q+)Γg5(q;P)S(q−). (26)
dependenceofr(ζ) ispreciselythatrequiredtoguarantee Zq
H
the r.h.s. of Eq. (18) is independent of the renormalisa- Equation (16) can be rewritten in terms of the fully-
tion point. (NB. r(ζ) is finite, and Eq. (18) valid, for amputated quark-antiquark scattering amplitude: M =
H
arbitrary values of the current-quark masses [19, 20].) K + K(SS)K + ..., and in the neighbourhood of the
In the chiral limit the existence of a solution of Eq. lowest mass pole
(9) with B (p2) 6= 0; i.e., DCSB, necessarily entails [17]
0
1
that Eqs. (15), (16) exhibit a massless pole solution: the M(q,k;P)=Γ (q;P) Γ¯(k;−P)+R(q,k;P),
Goldstone mode, which is described by H P2+m2H
(27)
Γg(k;P) = λgγ iE (k;P)+γ·PF (k;P) (21) where R is regular in this neighbourhood.
0 5 0 0 Assuming SU (3) flavour symmetry, substituting Eq.
f
(cid:20)
(27) into Eq. (26) gives
+ γ·kk·PG (k;P)+σ k P H (k;P) ,
0 µν µ ν 0
1
(cid:21) ωfg(P)=δfg Z−2r2 +... (28)
wherein fπ0E0(k;0) = B0(k2). (The index “0” indicates 5 P2+m2H m H
a quantity calculated in the chiral limit.) It follows im-
(the ellipsis denotes terms regular in the pole’s neigh-
mediately [17] that
bourhood). It follows that the large-x2 behaviour of
Λ
f0r(ζ) =−hq¯qi0 = lim Z (ζ,Λ)N tr S (q), (22) m2 ∆ (x) (29)
π 0 ζ 4 c D 0 bm 5
Λ→∞ Zq
is a measure of the renormalisation-group-invariant
where the trace is only over Dirac indices. This result
and multiplicative renormalisability entail 2
(ζ)
m(ζ)r (30)
H
hq¯qi0
ζ =Z (ζ,ζ′)Z−1(ζ,ζ′)=Z (ζ,ζ′), (23) h i
hq¯qi0 4 2 m that appears in Eq. (18). Hence the correlator in Eq.
ζ′
(25) providesadirectmeans ofestimating the OPEcon-
whereZmisthemass-renormalisationconstant. Itisthus densate in lattice simulations [21], one whose ultraviolet
apparentthatthechirallimitbehaviourofr(ζ) yieldsthe behaviour ensures a well-defined and calculable evolu-
H
OPE condensate evolved to a renormalisationpoint ζ. tionundertherenormalisationgroupforanyvalueofthe
It is important to recall that the DSEs reproduce ev- current-quark mass. (NB. f can similarly be extracted
π
ery diagram in perturbation theory. Therefore a weak from the axial-vector correlator analogous to Eq. (25).)
coupling expansion of Eq. (9) yields the perturbative se- ThemodelofRef.[22]yieldsamesonmasstrajectoryvia
riesforthe dressed-quarkpropagator. Thismaybe illus- Eq.(18) thatprovidesaqualitativeandquantitativeun-
tratedbytheresultforthescalarpieceofthepropagator derstanding of recent quenched lattice simulations [20].
4
III. BANKS-CASHER RELATION evaluated at a fixed value of the regularisationscale:
σ(m;ζ) := − limhq¯(x)q(0)i
A. Continuum analysis x→0
Λ
= Z (ζ,Λ)N tr S (p;ζ), (37)
It is readily apparent that Eq. (6) is meaningless as 4 c D m
Zp
written: dimensionalcountingrevealsther.h.s.hasmass-
dimensionthreeandsinceλwillatsomepointbegreater where the argument remains m = mbm(Λ), which
thananyrelevantinternalscale,theintegralmustdiverge is permitted because mbm(Λ) is proportional to the
as Λ2, where Λ is the regularising mass-scale. renormalisation-point-independent current-quark mass.
The renormalisation constant Z vanishes logarithmi-
Tolearnmore,considerthetraceoftheunrenormalised 4
cally with increasing Λ and hence one still has σ(m) ∼
massive dressed-quark propagator:
Λ2mbm(Λ). However, using Eq. (22) it is clear that for
Λ anyfinitebutlargevalueofΛandtoleranceδ,itisalways
σ˜(m):=NctrD S˜m(p), (31) possible to find mbm(Λ) such that
δ
Zp
σ(m;ζ)+hq¯qi0 <δ, ∀mbm <mbm. (38)
ζ δ
evaluated at a fixed value of the regularisation scale, Λ.
This Schwinger function can be identified with the l.h.s. This is true in QCD. It can be illustrated using the
ofEq.(6). Furthermore,assumethatσ˜(m)hasaspectral DSE model of Ref. [2], which preserves the one-loop
representation, since this is the essence of the Banks- renormalisation group properties of QCD. In Fig. 1 we
Casher relation: plot σ(m,ζ), evaluated using a hard cutoff, Λ, on the
integralinEq.(37),calculatedwiththe massivedressed-
Λ ρ˜(λ) quark propagatorsobtained by solving the gap equation
σ˜(m):=2m dλ , (32)
λ2+m2 asdescribedintheappendix. SinceEq.(38)specifiesthe
Z0
domain on which the value of σ(m;ζ) is determined by
where m=mbm(Λ). Equation (32) entails nonperturbative effects, one anticipates
mbm(Λ)≈−hq¯qi0/Λ2 ∼10−9 (39)
1 δ
ρ˜(λ)= lim [σ˜(iλ+η)−σ˜(iλ−η)] . (33)
2π η→0+ for Λ = 2.0TeV in QCD where |hq¯qi0| ∼ Λ3 , an esti-
QCD
mate confirmed in Fig. 1.
Thecontentandmeaningofthissequenceofequations The dotted line in Fig. 1 is
is well illustrated by inserting the free quark propagator
2 Λ
in Eq. (31). The integral thus obtained is readily evalu- σ(m,ζ)=−hq¯qi0 arctan
ated using dimensional regularisation: ζ π m
N
+ Z (ζ,Λ) c m Λ2−m2ln 1+Λ2/m2 .(40)
N m2 1 4 4π2
σ˜ (m)= c m3 ln + +γ−ln4π . (34)
free 4π2 ζ2 ε (NB. We used the on(cid:2)e-loop form(cid:0)ula: Z (ζ(cid:1),(cid:3)Λ) =
(cid:20) (cid:21) 4
[α(Λ)/α(ζ)]γm, for the numerical comparison.) The
With Eq.(33)the regularisationdependent terms cancel difference between Eq. (40) and the curve is of
and one obtains O(α(Λ)m(Λ)Λ2) because the DSE model incorporates
QCD’sone-loopbehaviour. InFig.1wealsoplotσ(m,ζ)
N
ρ˜(λ)= c λ3. (35) obtainedintheabsenceofconfinement,inwhichcase[24]
4π2
hq¯qi0 ≡0, as is apparent.
The one-loop contribution to ρ˜(λ) has been evaluated The discussion establishes that σ(m,ζ) has a regular
using the same procedure [23]. It is also proportional chiral limit in QCD and is a monotonically increasing
to λ3 and arises from the m3lnm2/ζ2 terms in σ˜(m). convex-upfunction. Itfollowsthatσ(m,ζ)hasaspectral
In fact, every term obtainable in perturbation theory is representation:
proportional to λ3, for precisely the same reason that Λ ρ(λ)
each term in the perturbative expression for the scalar σ(m,ζ)=2m . (41)
λ2+m2
part of the quark propagator is proportional to m, see Z0
Eq. (24). Hence, at every order in perturbation theory, This lays the vital plank in a veracious connection be-
tween the condensates in Eqs. (1) and (8). On the do-
ρ˜(λ=0)=0 (36) main specified by Eq. (39), the behaviour of σ(m,ζ) in
Eq.(37)is givenby Eq.(40), whichyields, viaEq.(33),
and h0|q¯q|0i = 0. A nonzero value of ρ(0) is plainly an
N
essentially nonperturbative effect. πρ(λ)=−hq¯qi0+Z (ζ,Λ) c λ3+... (42)
A precise analysis requires that attention be paid to ζ 4 4π
renormalisation. Consider then the gauge-parameter- where the ellipsis denotes contributions from the higher-
independent trace of the renormalised quark propagator order terms implicit in Eq. (40).
5
100
0.5
β=5.4
80
0.4
V)
Ge 0.3 60
1/3m) ( λ( )
σ( 0.2 ρ 40 β=5.6 β=5.8
0.1
20
0.0
10−10 10−9 10−8 10−7 10−6
m (Λ) (GeV)
bm 0
0 0.1 0.2
FIG. 1: Circles/solid-line: σ(m)1/3 in Eq. (37) as afunction λ
of the current-quark bare-mass, evaluated using the dressed-
quark propagator obtained in the model of Ref. [2]; dashed
FIG. 2: Spectral density of the staggered Dirac operator
line: themodel’s value of (−hq¯qi0ζ=1GeV)=(0.24GeV)3; and in quenched SU(3) gauge theory calculated on a 44-lattice.
dottedline: Eq.(40). Diamonds: σ(m)1/3evaluatedinanon- (Measuredinunitsofthelatticespacing.) Thedeconfinement
chqo¯qnifi0n≡ing0.versionofthemodel;dot-dashedline: Eq.(40)with transition takes place at β >∼5.6.
justenteredthedeconfineddomainandclosetothetran-
B. Comparison with a lattice-QCD simulation sitionboundarynonperturbativeeffectsarestillmaterial,
as seen, e.g., in the heavy-quark potential and equation
In Fig. 2 we plot the spectral density of the staggered ofstate[25]. It isa modernchallengeto determine those
Dirac operator in quenched SU(3) gauge theory calcu- gaugecouplingsandlatticeparametersforwhichthedata
lated with 3000 configurations obtained on a V = 44- are quantitatively consistent with Eq. (42).
lattice,inthevicinityofthedeconfiningphasetransition
at β >∼ 5.6. Details of the simulation are given in Ref.
[12]. Dimensioned quantities are measured in units of IV. EPILOGUE
1/a,where a is the lattice spacing,and it is ρ(λ)/V that
shouldbecomparedwiththecontinuumspectraldensity. We verified that the gauge-invarianttrace of the mas-
While the effect of finite lattice volume is apparent in sivedressed-quarkpropagatorpossessesaspectralrepre-
Fig. 2 for λa>∼0.1, the behaviour at smallλa is qualita- sentation when considered as a function of the current-
tively in agreement with Eq. (42) and Fig. 1: a nonzero quark mass. This is key to establishing that the OPE
OPE condensate dominates the Dirac spectrum in the condensate, which sets the ultraviolet scale for the
confined domain; and it vanishes in the deconfined do- momentum-dependenceofthe traceofthe dressed-quark
main whereupon ρ(0) = 0 and the perturbative evolu- propagator, does indeed measure the density of far-
tion, Eq. (35), is manifest. infraredeigenvaluesofthegauge-averagedmasslessDirac
To be more quantitative, we note that at β = 5.4, operator, a` la the Banks-Casher relation. This relation
ρ(0)a3 ≈70, so that is intuitively appealing because a measurable accumu-
lation of eigenvalues of the massless Dirac operator at
πρ(0)a3/V ≈(0.95)3. (43) zero-virtuality expresses a mass gap in its spectrum.
In practice, there are three main parameters in a sim-
ThevalueofthelatticespacingwasnotmeasuredinRef.
ulation of lattice-QCD: the lattice volume, characterised
[12] but one can nevertheless assess the scale of Eq. (43)
by a length L; the lattice spacing, a; and the current-
by supposing a ∼ 0.3fm ∼ 0.3/Λ , a value typical of
QCD quark mass, m. So long as the lattice size is large
small couplings, β, wherewith the r.h.s. is ∼ (3Λ )3.
QCD comparedwiththecurrent-quark’sComptonwavelength;
Thisistoolargebutnotunreasonablegiventheparame-
viz.,L≫1/m,thendynamicalchiralsymmetrybreaking
tersofthesimulation,itserrorsandthesystematicuncer-
canbeexpressedinthesimulation. Supposingthattobe
tainties in our estimate. One can alsofit the lattice data
the case then, as we have explicated, so long as the lat-
at β = 5.8, whereby one finds ρ(λ) ∝ λ3 on λ < 0.1 but
tice spacing is small compared with the current-quark’s
with a proportionality constant larger than that antici-
Compton wavelength; i.e., a≪1/m≪L,
patedfromperturbationtheory;viz.Eq.(42). Somemis-
match is to be expected because at β =5.8 one has only πρ(0)≈−hq¯qi0 , (44)
1/a
6
where the r.h.s. is the scale-dependent OPE condensate The ultraviolet (Q2 >∼ 1GeV2) behaviour of G(Q2) in
[ζ =Λ=1/a in Eq. (42)]. Eq. (A1) is fixed by the known behaviour of the quark-
In our continuum analysis we found that one requires antiquark scattering kernel [2]. The form of that kernel
am<∼(aΛQCD)3 ifρ(λ=0)istoprovideaveraciousesti- ontheinfrareddomainiscurrentlyunknownandamodel
mate of the OPE condensate. The residue at the lowest- is employed to complete the specification of the kernel.
masspole in the flavour-nonsingletpseudoscalarvacuum An efficacious form is [2]
polarisation provides a measure of the OPE condensate
that is accurate for larger current-quark masses. G(Q2) =8π4Dδ4(k)+ 4π2DQ2e−Q2/ω2
Q2 ω6
γ π
+4π m F(Q2), (A2)
Acknowledgments 2
1ln τ + 1+Q2/Λ2
2 QCD
We benefited from interactions with M.A. Pichowsky (cid:20) (cid:16) (cid:17) (cid:21)
and P.C. Tandy. This work was supported by: Deutsche where: F(Q2) = [1 − exp(−Q2/[4m2])]/Q2, m =
t t
Forschungsgemeinschaft, under contract no. Ro 1146/3- 0.5GeV; τ = e2−1; γ = 12/(33−2N ), N = 4; and
m f f
1; the Department of Energy, Nuclear Physics Division, Λ = Λ(4) = 0.234GeV. The true parameters in Eq.
QCD
under contract no. W-31-109-ENG-38; the National Sci- MS
(A2) are D and ω, however, they are not independent:
ence Foundationunder grantno.INT-0129236;andben-
in fitting, a change in one is compensated by altering
efitedfromtheresourcesoftheNationalEnergyResearch
the other, with fitted observables changing little along a
Scientific Computing Center.
trajectory ωD =(0.6GeV)3. Herein we used
D =(0.884GeV)2, ω =0.3GeV. (A3)
APPENDIX A: MODEL GAP EQUATION
A non-confining model is obtained with D =0.
Thegapequation’skernelisbuiltfromaproductofthe Equation (A1), is readily solved for the dressed-quark
dressed-gluon propagator and dressed-quark-gluon ver- propagator,with the renormalisationconstants fixed via
tex. Itcanbecalculatedinperturbationtheorybutthat Eq.(11). Thatsolutionprovidestheelementsusedinthe
is inadequate for the study of intrinsically nonperturba- illustration of Sec. III.
tivephenomena. Tomakemodel-independentstatements
about DCSB one must employ an alternative systematic
and chiral symmetry preserving truncation scheme.
The leading order term in one such scheme [26] is
the renormalisation-group-improved rainbow truncation
of the gap equation (Q=p−q):
S(p)−1 =Z (iγ·p+mbm)
2
Λ λa λa
+ G(Q2)Dfree(Q) γ S(q) γ . (A1)
µν 2 µ 2 ν
Zq
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